Question

Part A: Suppose a random variable X have mean of µ and standard deviation σ. Let a and b be constants.

i) Derive the expected value of aX + b.

ii) Derive standard deviation of aX + b

Part B: Suppose that in country A, the price of certain good has a mean of $100 and a variance of 25, in A-dollars. Country B has a fixed exchange rate with A so that it takes 2 B-dollars to buy 1 A-dollar. What is the expected price of this good in B-dollars? What is its variance in B-dollars? What are the expected price and variance if the exchange rate were three-to-one?

Answer 1

A )   We have a R.V X with mean   and standarad deviation

a and b are some fixed constants

NOTE :

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( i ) Expectation of ( aX + b )

  

( ii ) Standard Deviation of ( aX + b )

B )  Country A :

• Mean Price of goods is 100 A dollars  
• Variance is 25 A dollars  

Country B :

• Fixed exchange rate with A so it takes 2 B dollars to buy 1 A dollar

Taking Expectation on both sides :

Taking Variance on both sides :

Now , Exchance rate is three - to - one i.e. ,  it takes 3 B dollars to buy 1 A dollar

Taking Expectation on both sides :

Taking Variance on both sides :

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